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- W4289305130 abstract "We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where $0<q<1$. Consequently, we can sample random spanning forests in a graph and (approximately) compute the reliability polynomial of any matroid. We also prove the thirty year old conjecture of Mihail and Vazirani that the bases exchange graph of any matroid has expansion at least 1. One of our key observations is a close connection between pure simplicial complexes and multiaffine homogeneous polynomials. Specifically, if $X$ is a pure simplicial complex with positive weights on its maximal faces, we can associate with $X$ a multiaffine homogeneous polynomial $p_{X}$ such that the eigenvalues of the localized random walks on $X$ correspond to the eigenvalues of the Hessian of derivatives of $p_{X}$." @default.
- W4289305130 created "2022-08-02" @default.
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- W4289305130 date "2018-11-05" @default.
- W4289305130 modified "2023-09-28" @default.
- W4289305130 title "Log-Concave Polynomials II: High-Dimensional Walks and an FPRAS for Counting Bases of a Matroid" @default.
- W4289305130 doi "https://doi.org/10.48550/arxiv.1811.01816" @default.
- W4289305130 hasPublicationYear "2018" @default.
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